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Theorem hmphtop 17485
Description: Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmphtop  |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top ) )

Proof of Theorem hmphtop
StepHypRef Expression
1 df-hmph 17463 . . 3  |-  ~=  =  ( `'  Homeo  " ( _V  \  1o ) )
2 cnvimass 5049 . . . 4  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 17464 . . . . 5  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5359 . . . . 5  |-  (  Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 8 . . . 4  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3223 . . 3  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3221 . 2  |-  ~=  C_  ( Top  X.  Top )
87brel 4753 1  |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   class class class wbr 4039    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708    Fn wfn 5266   1oc1o 6488   Topctop 16647    Homeo chmeo 17460    ~= chmph 17461
This theorem is referenced by:  hmphtop1  17486  hmphtop2  17487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-hmeo 17462  df-hmph 17463
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